Answer

Verified

383.6k+ views

Hint: First, analyse the odd and even terms separately from an A.P with common difference, from that of the original A.P. Compute the sum of odd and even terms and find the ratio in terms of n.

Complete step-by-step answer:

The A.P. containing 2n + 1 terms can be divided into two different A.P.s by separating the odd and even terms of the original A.P. The A.P. formed by taking the odd terms will have (n + 1) terms and the A.P. formed by taking the even terms will have n terms.

Let the first term of the original A.P. be a and the common difference be d. Then, the first odd term is a and the first even term is a + d.

We know the formula for the \[{n^{th}}\] term of the A.P.

\[{t_n} = a + (n - 1)d........(1)\]

The last even term of the A.P. is the \[2{n^{th}}\] term and it is given by:

\[{l_{even}} = a + (2n - 1)d.........(2)\]

The last odd term of the A.P. is the \[{(2n + 1)^{th}}\] term and it is given by:

\[{l_{odd}} = a + (2n + 1 - 1)d\]

\[{l_{odd}} = a + 2nd...........(3)\]

We know that the sum to n terms of the A.P. is given as:

\[{S_n} = \dfrac{n}{2}(a + l)............(4)\]

The sum of odd terms of the A.P. is given as:

\[{S_{odd}} = {S_{n + 1}} = \dfrac{{n + 1}}{2}(a + {l_{odd}})\]

From equation (3), we get:

\[{S_{odd}} = \dfrac{{n + 1}}{2}(a + a + 2nd)\]

\[{S_{odd}} = \dfrac{{n + 1}}{2}(2a + 2nd)\]

Taking 2 out as common and cancelling with the denominator, we have:

\[{S_{odd}} = (n + 1)(a + nd).........(5)\]

The sum of even terms of the A.P. is given as:

\[{S_{even}} = {S_n} = \dfrac{n}{2}(a + d + {l_{even}})\]

Substituting equation (2) in the above equation, we have:

\[{S_{even}} = \dfrac{n}{2}(a + d + a + (2n - 1)d)\]

\[{S_{even}} = \dfrac{n}{2}(2a + 2nd)\]

Taking 2 out as common and cancelling with the denominator, we have:

\[{S_{even}} = n(a + nd).........(6)\]

Dividing equation (5) by equation (6), we have:

\[\dfrac{{{S_{odd}}}}{{{S_{even}}}} = \dfrac{{(n + 1)(a + nd)}}{{n(a + nd)}}\]

Cancelling (a + nd), we have:

\[\dfrac{{{S_{odd}}}}{{{S_{even}}}} = \dfrac{{n + 1}}{n}\]

Hence, we proved that the ratio of the sum of odd terms and the sum of even terms is (n + 1):n

Note: You must be careful when you choose the first even terms of the A.P, it is a + d and not a. The number of odd terms is (n + 1) and not n.

Complete step-by-step answer:

The A.P. containing 2n + 1 terms can be divided into two different A.P.s by separating the odd and even terms of the original A.P. The A.P. formed by taking the odd terms will have (n + 1) terms and the A.P. formed by taking the even terms will have n terms.

Let the first term of the original A.P. be a and the common difference be d. Then, the first odd term is a and the first even term is a + d.

We know the formula for the \[{n^{th}}\] term of the A.P.

\[{t_n} = a + (n - 1)d........(1)\]

The last even term of the A.P. is the \[2{n^{th}}\] term and it is given by:

\[{l_{even}} = a + (2n - 1)d.........(2)\]

The last odd term of the A.P. is the \[{(2n + 1)^{th}}\] term and it is given by:

\[{l_{odd}} = a + (2n + 1 - 1)d\]

\[{l_{odd}} = a + 2nd...........(3)\]

We know that the sum to n terms of the A.P. is given as:

\[{S_n} = \dfrac{n}{2}(a + l)............(4)\]

The sum of odd terms of the A.P. is given as:

\[{S_{odd}} = {S_{n + 1}} = \dfrac{{n + 1}}{2}(a + {l_{odd}})\]

From equation (3), we get:

\[{S_{odd}} = \dfrac{{n + 1}}{2}(a + a + 2nd)\]

\[{S_{odd}} = \dfrac{{n + 1}}{2}(2a + 2nd)\]

Taking 2 out as common and cancelling with the denominator, we have:

\[{S_{odd}} = (n + 1)(a + nd).........(5)\]

The sum of even terms of the A.P. is given as:

\[{S_{even}} = {S_n} = \dfrac{n}{2}(a + d + {l_{even}})\]

Substituting equation (2) in the above equation, we have:

\[{S_{even}} = \dfrac{n}{2}(a + d + a + (2n - 1)d)\]

\[{S_{even}} = \dfrac{n}{2}(2a + 2nd)\]

Taking 2 out as common and cancelling with the denominator, we have:

\[{S_{even}} = n(a + nd).........(6)\]

Dividing equation (5) by equation (6), we have:

\[\dfrac{{{S_{odd}}}}{{{S_{even}}}} = \dfrac{{(n + 1)(a + nd)}}{{n(a + nd)}}\]

Cancelling (a + nd), we have:

\[\dfrac{{{S_{odd}}}}{{{S_{even}}}} = \dfrac{{n + 1}}{n}\]

Hence, we proved that the ratio of the sum of odd terms and the sum of even terms is (n + 1):n

Note: You must be careful when you choose the first even terms of the A.P, it is a + d and not a. The number of odd terms is (n + 1) and not n.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

A rainbow has circular shape because A The earth is class 11 physics CBSE

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell